The total number of roots of the equation `| x-x^2-1|=|2x - 3-x^2|` is
A. o
B. 1
C. 2
D. infinity many

Options

A. 0
B. 1
C. 2
D. infinity many

Answer

To solve the equation \( |x - x^2 - 1| = |2x - 3 - x^2| \), we will analyze the absolute values by considering different cases based on the expressions inside the absolute values.

### Step 1: Identify the expressions
The equation can be rewritten as:
1. \( A = x - x^2 - 1 \)
2. \( B = 2x - 3 - x^2 \)

We need to solve \( |A| = |B| \).

### Step 2: Consider the cases for absolute values
We have two cases to consider:
1. Case 1: \( A = B \)
2. Case 2: \( A = -B \)

### Case 1: \( A = B \)
Setting \( A = B \):
\[ x - x^2 - 1 = 2x - 3 - x^2 \]
Simplifying this:
\[ x - x^2 - 1 = 2x - 3 - x^2 \]
Cancelling \( -x^2 \) from both sides:
\[ x - 1 = 2x - 3 \]
Rearranging gives:
\[ -1 + 3 = 2x - x \]
\[ 2 = x \]

### Case 2: \( A = -B \)
Setting \( A = -B \):
\[ x - x^2 - 1 = - (2x - 3 - x^2) \]
This simplifies to:
\[ x - x^2 - 1 = -2x + 3 + x^2 \]
Rearranging gives:
\[ x - x^2 + 2x - 3 - x^2 - 1 = 0 \]
Combining like terms:
\[ -2x^2 + 3x - 4 = 0 \]
Multiplying through by -1:
\[ 2x^2 - 3x + 4 = 0 \]

### Step 3: Calculate the discriminant
To find the number of solutions, we calculate the discriminant \( D \) of the quadratic equation \( 2x^2 - 3x + 4 = 0 \):
\[ D = b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot 4 = 9 - 32 = -23 \]
Since the discriminant is negative, this quadratic has no real roots.

### Step 4: Summary of solutions
From Case 1, we found one solution \( x = 2 \). From Case 2, we found that there are no additional solutions due to the negative discriminant.

### Conclusion
The total number of roots of the equation \( |x - x^2 - 1| = |2x - 3 - x^2| \) is **1**.

### Answer
The correct option is **B. 1**.

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